hwk1-sol - Due date: June 19, 2009, Friday Math 116...

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Due date: June 19, 2009, Friday Math 116 Calculus – Homework # 1 – Solutions Remark: Solve all the problems. One of the problems, randomly chosen, will be graded. Q-1) Can the function f ( x,y ) = sin x sin 3 y 1 - cos( x 2 + y 2 ) be defined at the origin in such a way that it becomes continuous there? If so, how? Solution: Limit of the expression as ( x,y ) goes to (0 , 0) along x -axis (or y -axis) is zero but limit along y = x direction is 1 / 2. So this function cannot be continuously extended to the origin. Q-2) Directional derivative of a function f ( x,y ) at the point ( - 1 , 1) in the direction ~ i + ~ j is 5, and in the direction ~ i - ~ j is - 5. Find the directional derivative of f ( x,y ) at ( - 1 , 1) in the direction of - ~ i - 2 ~ j . Solution: ~u = ( 1 2 , 1 2 ), ~v = ( 1 2 , - 1 2 ) and ~w = ( - 1 5 , - 2 5 ) are the unit vectors in the directions of ~ i + ~ j , ~ i - ~ j and - ~ i - 2 ~ j respectively. We are given that
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hwk1-sol - Due date: June 19, 2009, Friday Math 116...

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